## Bangalore workshop [ಬೆಂಗಳೂರು ಕಾರ್ಯಾಗಾರ]

**S**econd day at the Indo-French Centre for Applied Mathematics and the workshop. Maybe not the most exciting day in terms of talks (as I missed the first two plenary sessions by (a) oversleeping and (b) running across the campus!). However I had a neat talk with another conference participant that led to [what I think are] interesting questions… (And a very good meal in a local restaurant as the guest house had not booked me for dinner!)

**T**o wit: given a target like

the simulation of λ can be demarginalised into the simulation of

where **z** is a latent (and artificial) variable. This means a Gibbs sampler simulating λ given z and z given λ can produce an outcome from the target (*). Interestingly, another completion is to consider that the z_{i}‘s are U(0,y_{i}) and to see the quantity

as an unbiased estimator of the target. What’s quite intriguing is that the quantity remains the same but with different motivations: (a) demarginalisation versus unbiasedness and (b) z_{i} ∼ Exp(λ) versus z_{i} ∼ U(0,y_{i}). The stationary is the same, as shown by the graph below, the core distributions are [formally] the same, … but the reasoning deeply differs.

**O**bviously, since unbiased estimators of the likelihood can be justified by auxiliary variable arguments, this is not in fine a big surprise. Still, I had not thought of the analogy between demarginalisation and unbiased likelihood estimation previously.**H**ere are the R procedures if you are interested:

n=29 y=rexp(n) T=10^5 #MCMC.1 lam=rep(1,T) z=runif(n)*y for (t in 1:T){ lam[t]=rgamma(1,shap=2,rate=1+sum(z)) z=-log(1-runif(n)*(1-exp(-lam[t]*y)))/lam[t] } #MCMC.2 fam=rep(1,T) z=runif(n)*y for (t in 1:T){ fam[t]=rgamma(1,shap=2,rate=1+sum(z)) z=runif(n)*y }

July 31, 2014 at 8:58 am

I thought this was Nicolas Chopin’s main thrust when talking about these pseudo-marginal algorithms: unbiasedness is a consequence, but not the important thing. The expansion/re-marginalisation is the easiest way to work with these things (because just having unbiasedness isn’t enough – you’ve got to propagate the estimates forward correctly in the accept/reject step)